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Dynamics of Microscale Liquid Propagation in Micropillar Arrays Mohamed H Alhosani, and TieJun Zhang Langmuir, Just Accepted Manuscript • Publication Date (Web): 31 May 2017 Downloaded from http://pubs.acs.org on June 6, 2017
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Dynamics of Microscale Liquid Propagation in Micropillar Arrays Mohamed H. Alhosani, TieJun Zhang* Department of Mechanical and Materials Engineering, Masdar Institute, Khalifa University of Science and Technology, P.O. Box 54224, Abu Dhabi, UAE *Email:
[email protected] ABSTRACT
Understanding the dynamics of microscale liquid propagation in micropillar arrays can lead to significant enhancement in macroscopic propagation modeling. Such phenomenon is fairly complicated and fundamental understanding is lacking. The aim here is to estimate three main parameters in liquid propagation, capillary pressure, average liquid height and contact angle on pillar side, through modeling and experimental validation. We show that the capillary pressure is not constant during liquid propagation, and average capillary pressure is evaluated by using its maximum and minimum values. Average liquid height influences the permeability of such structure, which is challenging to determine due to complicated 3-D meniscus shape. A simple physical model is provided in this paper to predict average liquid height with less than 7% error. Contact angle on micropillar side, which have considerable impact on the capillary pressure and the average liquid height, have been debated for long time. We propose a model to predict this contact angle and validate it against experimental values in open literature. Our findings also 1 ACS Paragon Plus Environment
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indicate that the microscopic motion of liquid front is significantly affected by the ratio of pillar height to edge-to-edge spacing, and a correlation is provided for quantification. The proposed models are able to predict the droplet spreading dynamics and estimate spreading distance and time reasonably.
KEYWORDS: Capillary pressure, Permeability, Contact angle, Wicking, Liquid propagation, Dynamic, Droplet spreading
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1. INTRODUCTION Liquid propagation (wicking) in porous media have been extensively studied owing to its application in variety of engineering applications such as thermal management of high heat flux devices, microfluidics and power generation. The advantage of such surfaces is that capillary pressure provides effective liquid transport mechanism for small devices. These surfaces can be produced by introducing micro/nanostructures on intrinsically hydrophilic surfaces based on Wenzel equation1. Among these surfaces, silicon micropillar arrays have been extensively studied to understand the fundamentals of liquid propagation2–7. Using photolithography technique and deep reactive ion itching (DRIE), precise pillar diameter, spacing and height can be manufactured which is crucial for fundamental understanding. Also, scalable and low-cost micro/nanostructured surface can be fabricated on metallic surfaces using different methods such as anodization8,9, oxidation3, electrochemical deposition3,8 and biotemplated nanofabrication10. One of the early studies in liquid propagation was done by Washburn11 to estimate liquid rise rate in capillary tubes. The model was obtained by balancing capillary pressure with viscous resistance, which was then extended to predict liquid propagation in porous media. The main challenge in such a problem is to estimate capillary pressure and viscous resistance (permeability). Recently, Xiao at al.12 proposed semianalytical model to predict liquid propagation rate in micropillar array. The capillary pressure was determined by predicting the meniscus shape using energy minimization software called “surface evolver” and validated using interferometry (confocal microscope images). Brinkman’s equation was solved numerically to obtain permeability. The model provides good predictions when the height to spacing ratio is larger than 1 and diameter to spacing ratio is lower than 0.57 and over-predicts outside this range. To overcome this problem, the same group13 studied the microscale liquid propagation dynamics 3 ACS Paragon Plus Environment
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in which the liquid propagation between two pillars (sweeping) was analyzed. The results show the importance of accounting microscale dynamics, especially in sparse structures. They also show that the sweeping distance scales with one-fifth power of time. Another study done by Tanner suggests that the sweeping distance scales with one-seventh power of time14. M. Rahman et al.10 studied the effect of liquid propagation rate on critical/dry-out heat flux. Propagation rate was measured with high speed imaging of liquid drawn from a 500µm capillary tube when in contact with the surface. The results show that liquid propagation rate is the single main factor that determines the dry-out heat flux on structured superhydrophilic surfaces. Optimization of liquid propagation on square micropillar arrays was done by Hale et al.15 based on several models and numerical simulations. It was found that for a given aspect ratio, there exists an optimum pillar spacing that provides the maximum flow rate. Another group also presented the dynamics of liquid propagation in nanopillars16 and reported that pillar structures have significant influence on the propagation process. The modeling results were then validated with experimental data from literature. Antao et al.17 provided simple yet accurate capillary pressure model based on the force balance in a unit cell of micropillar array. The model was used to estimate the maximum propagation rate, which is also a representation of dryout heat flux on such surfaces18. Despite the huge effort directed to understand the liquid propagation in micro/nanostructured surfaces, this phenomenon is still unclear and different models produce various results as shown by Horner et al.4 and Ravi et al.19,20. For instance, microscopic sweeping process is still vague and physical insight of this process can lead to significant improvement in macroscopic behavior. Also, an important parameter to understand liquid propagation is the contact angle on the pillar side, which was determined by interferometry by Antao et al.17 for water on silicon to be much 4 ACS Paragon Plus Environment
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larger than equilibrium contact angle. Also, Ravi et al.19,20 used a contact angle of 56° when simulating water propagation on silicon micropillar array. On the other hand, Zhu et al.18 used a contact angle of 30°. This shows that a model that can predict contact angle of liquid on micropillar side is lacking in literature. Another important parameter that affect liquid propagation rate is the permeability. One of the early studies to estimate permeability of micropillar array is done by Sangani and Acrivos in 19822. The proposed numerical model assumes flat meniscus (contact angle = 90°) between the pillars which lead to over-prediction of permeability since the flow area is larger. The effect of meniscus curvature was taken into account on Byon and Kim3 permeability model in which the surface energy minimization tool was used to obtain general correlation for average meniscus height. One main drawback is that this correlation can be significantly off if the parameters used is not within the range of the correlation. Recently, Zhu, et al.18 proposed that the meniscus height/curvature is changing from the liquid supply point along the wicking direction. Basically, the pressure difference in wicking system is due to curvature difference. This change in curvature results in different meniscus heights along the flow. Therefore, a simple universal model is needed to accurately predict average (effective) meniscus height within a unit cell as a function of pillar diameter, spacing, height and contact angle. In this paper, we show that capillary pressure during propagation and sweeping is not constant and therefore, average capillary pressure is proposed based on maximum and minimum capillary pressure. Also, a model that can predict contact angle on micropillar sidewall was produced, and validated against interferometry measurements from literature. Furthermore, we provide a simple yet accurate model to estimate average liquid height in a unit cell which is crucial for permeability estimation. Finally, average capillary pressure, average liquid height and contact 5 ACS Paragon Plus Environment
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angle models were validated using liquid rise and droplet spreading experiments on silicon micropillar arrays.
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2. RESULTS AND DISCUSSION In this work, we study the liquid propagation in hydrophilic micropillar arrays. A scanning electron microscope (SEM) image of such sample with pillar diameter of d, center-to-center spacing (pitch) of p and pillar height of h is shown in Figure 1a. The aim is to estimate the spreading time and spreading diameter when a liquid droplet is deposited in such surfaces (Figure 1b), also to predict the liquid propagation rate when such surface comes in contact with liquid reservoir as shown in Figure 1c.
a)
b)
c)
Figure 1. a) SEM image of fabricated sample with diameter d, pitch p and height h, b) schematics of droplet spreading in micropillar arrays, c) side view schematics of liquid propagation in micropillar arrays. Darcy’s law for flow in porous media can be used to simulate this problem,
(1)
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where, is liquid mass flow rate, ρ is liquid density, K is permeability, A is liquid flow area and µ is liquid viscosity. The driving pressure in this case is the capillary pressure (Pcap),
( ) @ ( )@
(2)
where L is the propagating distance from liquid reservoir. To estimate liquid propagation rate, we need to get the permeability and capillary pressure. 3.1 Average capillary pressure in micropillar arrays When liquid is moving from one row of pillars to the next one, the capillary pressure is not constant. For instance, the maximum capillary pressure is just after the liquid wets the micropillars, and the minimum capillary pressure (Pmin) is just before the liquid wets the next row of pillars. Applying the force balance in a unit cell of micropillar arrays suggests that capillary pressure follows the equation below (see Supporting Info. S-1 for derivation) 2 !"#$ % !&'"#$ ()
) %( 2 + ) 8
(3)
where θadv is the advancing contact angle and x is the distance from the pillar edge to the liquid front. Note that pressure difference (∆P) in Figure 2 is equals to Pcap. Due to DRIE process, scallop like structures are formed on the pillar side, which should results in a lower contact angle based on Wenzel equation. However, high-resolution SEM images show that pillar side roughness is equal to 1.01 – 1.02, which can be neglected (see Supporting Info. S-1 for image). Other studies also show similar results such as Ayon et al.21. However, this roughness can be changed by changing DRIE process parameters.
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a)
b)
Figure 2. a) Side view, b) top view of liquid front and forces on the meniscus during sweeping process. We define Pmax when the liquid front at 0 and Pmin when % as shown in Figure 3.
a)
b)
Figure 3.Top view of liquid front location when capillary pressure equal to a) Pmin, b) Pmax. Substituting the value of x in Eq. (3), Pmin and Pmax can be calculated as shown in Eq. (4) and Eq. (5), respectively. ,-
2 !"#$ % !&'"#$
) % .% 2 / 8
(4)
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!"#$ % !&'"#$ 2
) %( 2) 8
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(5)
The above model does not takes into account the effect of pillar height on capillary pressure because it did not appear in the force balance directly. Previous studies reported over-estimation of capillary pressure when short pillars are modeled13. The main parameter here is the ratio of pillar height to edge-to-edge spacing (h/(p-d)). When h/(p-d) < 1, the capillary pressure is overestimated. The reason is that the contact angle on pillar side (θpillar) is not the same as θadv for short pillars. Basically, for very short pillars, it is physically impossible to have θadv on both the bottom surface and pillar side as shown in Figure 4.
Figure 4. Liquid meniscus and contact angle on long and short pillars. To estimate θpillar from h/(p-d), two boundary conditions are used. If h = (p-d), both bottom surface and pillar side contact angle will equal to θadv. On the other hand, if pillar height approaches zero, θpillar will approach (90+θadv). Note that contact angle on pillar wall can be larger than θadv due to the edge effect on pillar top, which does not exist on the bottom surface as shown in Figure 4. Based on these two boundaries, an approximation is developed to estimate θpillar from h/(p-d) as shown below 10 ACS Paragon Plus Environment
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",00 90 2tan56(
ℎ ) + "#$ %
(6)
Since θpillar cannot be lower than θadv, max[θpillar, θadv] is substituted in the first term of Eq. (3) and Eq.(5). During the liquid propagation process, the average capillary pressure can be determined from the time average of the pressure along x. Previous study by Xiao et al.13 shows that the liquid front location (x) is a function of t1/5, where t is time. However, Tanner reported that x is function of t1/7. The first study which states sweeping distance scaling as t1/5 was done on silicon micropillar arrays. The second study on the other hand, was on smooth surface. In the case of silicon micropillar arrays, the meniscus is pinned at the pillar top with constant height, while in the second case the liquid thickness decreases. This implies that pillar height affect the sweeping distance scaling (β). Therefore, six samples are tested and the liquid (water) sweeping on plasma cleaned silicon micropillar arrays samples with h/(p-d) ranges between 0.5 to 1.6 are recorded with high speed camera at 5000 frames per second. It was found that h/(p-d) strongly affect β and a correlation is developed to estimate β from h/(p-d) as shown below. While this correlation is valid for the experimented range of h/(p-d), we do not expect β to keep increasing. There is a physical limitation in which β cannot go beyond that will be discussed later in this manuscript.
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0.6 Sweeping distance scaling (β)
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y = 0.2705x + 0.0784 R² = 0.9897
0.5 0.4 0.3 0.2 0.1 0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
h/(p-d)
Figure 5. Sweeping distance scaling (β) vs. h/(p-d). ℎ 8 0.2705 < = + 0.0784 %
(7)
Using this information, the average capillary pressure (Pavg) can be determined as shown below (see Supporting Info. S-3 for derivation) , E #
$? @
,6
B AB(&)C + D
(8)
% ) !",00 % !&'"#$ , A %(% ), D 2 2 8
The above model was for the edge unit cells. If we consider a unit cell with all 4 pillars wetted (center unit cell), the capillary pressure can be determined as that in17 (see Supporting Info. S-2 for more details) G-G
4 !" 4 % )
